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Studia Logica

, Volume 53, Issue 3, pp 389–396 | Cite as

A note on the ω-incompleteness formalization

  • Sergio Galvan
Article

Abstract

The paper studies two formal schemes related to ω-completeness.

LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where α(x) is a formula with the only free variable x): (a) (for anyn) (⊢ T α(n)), (b) ⊢ T x Pr T (α(x)) and (c) ⊢ T xα(x) (the notational conventions are those of Smoryński [3]). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) ⇒ (c) and (a) ⇒ (b), where α ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) ⇒ (c) is equivalent to ¬ConsT and 2. the schema corresponding to (a) ⇒ (b) is not consistent with 1-CONT. The former result follows from a simple adaptation of the ω-incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma.

Keywords

Mathematical Logic Formal Theory Base theoryS Computational Linguistic Formal Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    S. Galvan,Introduzione ai teoremi di incompletezza Angeli, Milano 1992.Google Scholar
  2. [2]
    J. Y. Girard,Proof Theory and Logical Complexity, Vol. I, Bibliopolis, Napoli 1987.Google Scholar
  3. [3]
    C. Smoryński,The Incompleteness Theorems, in J. Barwise (ed.),Handbook of Mathematical Logic North Holland, Amsterdam 1977, pp. 821–865.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Sergio Galvan
    • 1
  1. 1.Institute of PhilosophyUniversity of VeronaVeronaItaly

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